Overview
The binomial theorem expands (a + b)ⁿ without multiplying out every bracket.
The expansion
(a + b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ aⁿ⁻¹b + ⁿC₂ aⁿ⁻²b² + … + ⁿCₙ bⁿ
The coefficients are the binomial coefficients ⁿCᵣ (also Pascal's triangle).
General term
The term in bʳ is ⁿCᵣ aⁿ⁻ʳ bʳ — useful for finding one specific term without the whole expansion.
Worked example
(1 + x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴.
To find the x² term of (2 + x)⁵: ⁵C₂ × 2³ × x² = 10 × 8 × x² = 80x².
Common mistakes
- Forgetting to raise the whole term (e.g. (2)³, not just 2) to its power.
Exam tips
- The powers of a decrease and powers of b increase, always summing to n.